Rototranslational sum rules for nuclear dynamics via traveling pseudopotentials

We establish a set of exact sum rules that relate the interatomic force constants to the frequency-dependent electromagnetic susceptibility of a solid or molecule, thereby generalizing the long-established principles of rototranslational symmetry to the nonadiabatic regime. Crucially, we show that in practical numerical implementations these sum rules are violated, unless special precautions are taken in the treatment of the atomic pseudopotentials. We solve these issues once and for all by correctly adapting the pseudopotential to the motion of the corresponding nucleus, with a velocity dependence of the nonlocal operator. This prescription restores the correct Galilean covariance of the Schr\"odinger equation, and the expected identity between mechanical rototranslations and electromagnetic perturbations. These results conclusively fix a number of worrisome inconsistencies that were pointed out over the years in the context of linear-response theory restoring, e.g., the validity of the Larmor theorem, and the equivalence between the inertial and electrical definitions of the Drude weight in metals.